Local translation with pivot point established on the active object
Moving a multiple selection with the keystrokes 


gz global axis


is different than


gzz local axis


where the letter z refers to the z axis and can be replaced by any of the set {x, y, z, X, Y, Z}.  Feel free to experiment.

So having a selection of multiple objects move on their locally rotated axis is a feature.

You can [apply] a rotation.  And thus all objects would have the same rotational axis and different origins.

I do not know your long term goals for your model.  Tutorials may lead you to understand how to 


[join] objects
[parent] objects

Blender always has various ways to achieve your goal.


So what is written here is not a tutorial, and you may need to search further here at BSE.  Also pay attention to the additional information which appears in Blender 3D View at the bottom, when using either of the keystrokes above.

You may appear to be changing transformation axis although the video may be clipping the lower portion of the screen capture.  You can also change the pivot center.



I believe Blender is performing as intended.  As the user you should see this behavior as a good option and not something unintended.

In the image below the Blender User first grabbed an object.  Then pressed


F6


to see the other options of the grab/translate operator.



The famous website that rhymes with noonoob has tutorials on selection transformation in Blender.  Although I have not taken the time to find one that deeply addresses this topic you may find one.

https://www.youtube.com/watch?v=JaYhhaKDh5E

https://www.youtube.com/watch?v=wklRKGC7ai8